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We present a growth study and structural characterization of InP-InSb nanowire heterostructures. In contrast to planar epitaxy, this heterostructure can be realized in nanowires without the formation of dislocations, despite an extreme lattice-mismatch (10.4%). We obtain high crystal quality in the InSb nanowires, confirmed by a narrow 111 reflection peak measured by XRD. Additionally, the diameter dependence of the nanowire growth rate was investigated. An original competition between surface...[Show more] growth and nanowire growth is found, which can be controlled by varying the nanowire surface coverage. Finally, HRTEM and X-EDS investigations reveal that the InSb nanowire is always defect-free zinc-blende, and that the InP-InSb heterointerface is free from misfit dislocations, although single twin planes are common.